The reduction method in the theory of Lie ...

The reduction method in the theory of Lie-algebraically integrable oscillatory Hamiltonian systems. A.K. Prykarpatsky

Dept. of Appl. Mathem. at AGH, Krakow 30059, POLAND, Dept. of Physics at EMU of Gazimagusa, N. Cyprus and Dept. of Nonlinear Math. Analysis at IAPPM of the NAS, Lviv 290601 UKRAINA

0. Introduction.

As is well known [1,4] symmetry analysis of nonlinear dynamical systems on a smooth manifold M gives rise in many cases to exhibiting its many hidden but interesting properties, in particular such as being integrable by quadratures due to the Liouville-Arnold theorem [2]. In case when the manifold M can be represented as the cotangent space T  (K ) to some subgroup K of a Lie group G naturally acting on it, the study of the corresponding ow can be recast via the reduction method [3] into the Hamiltonian framework due to the existence on T  (K ) the canonical Poisson structure. Furthermore, if the symmetry group G naturally generalizes to the loop group G+ () over  2 D0   () provides us C , then the corresponding momentum mapping l : T  (K ) ! G+ with a Lax type representation and related with it a complete set of commuting invariants . Such a scheme appeared to be very useful when proving the Liouville integrability of many nite-dimensional systems such as Kowalevskaya's top [3], Neumann type systems [5,6] and other. Below we study complete integrability of nonlinear oscillatory dynamical systems connected in particular both with the Cartan decomposition of a Lie algebra G = K+P , where K is the Lie algebra of a xed subgroup K  G with respect to an involution  : G ! G on the Lie group G, and with a Poisson action of special type on a symplectic matrix manifold.

1.Integrable systems on T  (K ) :the general scheme. Consider a Lie group G and an involution  on G: If K  G is its xed subgroup, then the Lie algebra G of the Lie group G admits the Cartan decomposition G = K+P with the induced involution mapping  = id on K and  = ,id on P . Denote also G  = K +P  the dual decomposition of the adjoint space G  : The cotangent space T  (K ) ' K  K by means of left translations on K: Assume now that the natural group action of G on T  (K ) is extended to that of the loop group G (),  2 D ; where D  C is a disc containing zero. Let G () be the Lie algebra of the loop group G () acting on the cotangent +

0

1

0

+

+

1

bundle T  (K ) ' K  K : If the action is Hamiltonian [1], one can de ne the corresponding momentum mapping l : T  (K ) ! G+ (). Here the adjoint space G+ () is de ned with respect to the following invariant and symmetric scalar product on G+ ():

<  (); () >,1= res2D0 1= <  (); () >G (1.1) for any  () 2 G+ (); () 2 G+ (), where <
;
>G denotes the standard Killing form on G . Any orbit passing through a point l(u; v; ) 2 G+ (); with (u; v) 2 T  (K ) being xed, is de ned naturally as Spanf PrG+ () (Adexp(,x()) l(u; v; ))g; (1.2) x()2G+ ()

where PrG+ () : G  (; ,1) ! G+ () denotes the projection upon G+ () parallelly to the subspace G, (); G  (; ,1) := G+ ()  G, ()Pwith G+ () = G, () and x() 2 G+ () is arbitrary element. Let G+ (P ) = f i2 + xi i : xi 2 G ; i  xi = (,1) xi for all i 2 Z+g; so G+ () = f j 2 + yj ,(j +1) : yj 2 G  ; yj = (,1)j +1 yj for all j 2 Z+g: Having for instance taken lb (u; v; ) = v(u) + ,1 b 2 G+ (); one can derive that lb (u; v; ) := Adexp(,x())lb (u; v; ) = Adu,1 v(u) + ,1Adu,1 b = (1.3)

Z Z

v(e) + ,1Adu,1 b for any (u; v) 2 K  K with u := exp x0 2 K and b 2 G  : Consider now an element a 2 G  (; ,1 ); where a 2 G  is constant. Since also (a; [G,(); G, ()]),1 = 0 = (a; G+ ()),1 (1.4) for any a 2 G  , we see that the element a 2 G  (; ,1 ) is an in nitesimal character of the Lie subalgebra G, (): Based now on the well known AdlerKostant-Symes (AKS) theorem [10], one can formulate the following theorem. (a;b) n (u; v; ) ), s; n 2 Theorem 1.1 All functional s;n (u; v) := res2D0 (s la;b Z, where

la;b (u; v; ) := lb (u; v; ) + a (1.5) are involutive on the cotangent space T  (K ) ' K  K with respect to the standard Poisson bracket on T  (K ): Since under the involution K 3 u :! u,1 2 K and Te (K ) 3 v(e) ! w(e) 2 T  (K ) ' K  combined with the permutation G  3 a ! b 2 G  the element la;b (u; v; ) ! lb;a (u; w; ); making it possible to rep) resent the ow on T  (K ) generated by the invariant 1(a;b ,n;n(u; v) 2 D(T  (K )); ) n 2 Z+ ; as the one generated by n(a;b ,3;;n(u; w): In case when a Lie algebra G is the Lie algebra of the connected subgroup G of SO(4; 3); the maximal compact subgroup K  G with the Lie algebra K 2

is isomorphic to so(4; 3):Thereby this pair (G ,K) can be used [11] for constructing integrable ows quadratic in momenta on T  (K ) , in particular the four -dimensional top and its generalizations. 2. Oscillatory dynamical systems on T  (K ) : an example. Consider now the case when a loop group G,() acts on T  (K ) ' K  K; where  2 D1 and D1  C is an open disc containing the in nite point. Put G,() the Lie algebra of the group G, () and G, () its adjoint space with respect to the scalar product <  (); () >0 := res2D1 < P (); () >G for any  () 2 G, () and () 2 G,(): AsPbefore, let G+ () = f i2 + xi i : xi 2 G , xi = (,1)i xi ; i 2 Z+g; G,() = f i2P+ yi ,(i+1) : yi 2 G , yi = (,1)i+1 yi ; i 2 Z+g:The adjoint space G+ () = f i2 + aii : ai 2 G  , xi = (,1)i xi; i 2 Z+g contains one-parametric orbits of the Ad - action, which can be interpreted as some nite-dimensional integrable Hamiltonian systems on T  (K ): For this to be a lot more clari ed, let us consider an element a2 2 G, () with a 2 P and calculate its orbit under the action Adexp(,x()) : G, () ! G, (); where x() 2 G, () is some element speci ed by a point (u; v) 2 T  (K ): We nd therefore that the orbit of the element a2 + b 2 G, () has the form:

ZZ

Z

la;b (u; v; ) = a2 + [x0; a] + [x1; a] + 1=2[x0; [x0; a]] + b; (2.1) in which one can make identi cations [x0; a] := q 2 Ka? and [x1; a] = p 2 Pa? with u := (exp x1) 2 K and b := [x0; a] 2 Ka  K due to the natural isomorphisms ad a : Ka? ! Pa? and ad a : Pa? ! Ka?  K: Similarly one can represent the forth element in (2.1) as (q) := PrPa 1=2[(ad a),1 q; q]; (2.2) where evidently  : Ka? ! Pa : Having assumed further that an element a 2 P is such that [Ga? ; Ga?]  Ga or equivalently G =Ga Ga? (the symmetric expansion), one easily veri es that [Pa? ; Ka?]  Pa ; or (q) = 1=2[(ad a),1q; q] since (q) 2 Pa for all q 2 Ka?: In virtue of the isomorphism between Pa? and Ka?; the orbit (2.1) evidently is dieomorphic both to Ka?  Pa? and to the cotangent space T  (Ka?): The space T  (Ka?) is endowed with the canonical Poissonian structure being equivalent to the standard Lie-Poisson structure upon the orbit (2.1): fqi ; qj g :=< l; [rqi(l); rqj (l) >0 = 0; (2.3) fqi; pj g :=< l; [rqi(l); rpj (l) >0 =< [fj ; ei ]; a >G ; fpi ; pj g :=< l; [rpi(l); rpj (l) >0 =< [fi ; fj ]; q >G ; for all i; j = 1; n and any (q; p) 2 T  (Ka? ); where r : D(T  (Ka? )) ! Ka  (Ka? )): When deriving (2.3) we denotes the usual gradient mapping on D(TP P made use of the following relationships: q := ni=1 qiei ; p := ni=1 pi fi ; where 3

fej = [fj ; a] 2 Ka? : j = 1; ng and ffj 2 Pa? : j = 1; ng are orthogonal bases in Ka? and Pa? correspondingly, that is < ei ; ej >G = ij =< fi ; fj >G for all

i; j = 1; n: As was mentioned in [12,13] the elements a 2 P satisfying the property G =Ga  Ga? can be found easily enough if one to consider a dual compact Lie algebra G =K iP . Then the Hermitian symmetric expansion G =Gia Gia? holds and the problem reduces to recounting all involutions  : G ! G in G commuting with the above Hermitian expansion and equal to " , id" upon the center of the Lie algebra Gia : The condition G =Ga  Ga? involved above on an element a 2 P implies obviously that Ga? = ad a(G ) = ad a(Ga? ), since by de nition ad a(Ga) = 0: Thus the element a 2 P de nes the projection operator Pa : G !G on G compatible with the involution  : G ! G , that is Pa  = Pa ; where Pa2 = Pa : The latter condition appears to be useful for practical calculations on which we shall not dwell here. To end this section, let us write down the corresponding Hamiltonian ows on T  (Ka?) in the componentwise form. The vector (q; p) 2 T  (Ka?) is a set of canonical coordinates on the orbit (2.1) since due to the imbedding [Pa? ; Pa?]  Ka , the bracket fpi ; pj g = 0 for all i; j = 1; n . As a result one obtains the following expression for the orbit point (2.1) : n n n X X X la;b (q; p; ) = a2 +  qi ei + ( pifi + 1=2 qi qj [ei; fj ] + b; (2.4) i=1

i=1

i;j =1

where in virtue of (2.3) fqi; qj g = 0 = fpi; pj g; fpi; qj g =< fj ; fi >G (2.5) for all i; j = 1; n: Evaluating the functional H = 1=2res2D1 ,1 hla;b (q; p; ); la;b(q; p; )iG on the orbit space T  (Ka? ) at b 2 Pa , one gets the Hamiltonian function n n X X H (q; p) = 1=2 p2j + 1=2 qiqj h[ei ; fj ]; biG + j =1

1= 8

n X n X i;j =1 s;l=1

i;j =1

qiqs < [ei ; fj ]; [es; fl ] >G qj ql ;

(2.6)

describing an unharmonic oscillatory dynamical system of particles on the axis R 3 qj ; j = 1; n; interacting with each other by means of a forth order potential. Based on theorem 1.1. one can formulate the following theorem.

Theorem 2.1. The unharmonic oscillatory dynamical system (2.6) on the orbit space T  (Ka?) with the Poisson brackets (2.5) is a completely LiouvilleArnold integrable [1,2,18] Hamiltonian system.

Choosing dierent semisimple Lie algebras G admitting the Hermitian symmetric expansion Ga  Ga? = G for some element a 2 P , where G = K  P is 4

the Cartan decomposition, one can build all of fourth order potential canonical Hamiltonian systems on T  (Ka?) ' T  (Rn) from [1].

3. Unharmonic oscillatory Hamiltonian systems and their Liealgebraic integrability. Consider now a dual matrix manifold M := Mn;  Mn; of dimension (n  2); n 2 Z ; endowed with the following natural symplectic structure ! = Sp(dQ| ^ dF ); (3.1) where (F; Q) 2 M and "Sp" means the standard trace operation. Let A () mean an analytical inside an open ring D 3 0 loop group acting on the manifold M as follows: for any (F; Q) 2 M and g() 2 A () 2

2

+

(2)

+

0

+

() F :g! Fg() := res2D0  ,1 Fg,1();

() (3.2) Q| :g! Q|g() := res2D0 g()Q|  ,1 ; where 2 Mn;n is some matrix whose spectrum ( )  D0 : Denote A+ () the Lie algebra of the Lie group A+ () , and put X A+ () = f aj j : aj 2 sl(2; R), j 2 Z+g: (3.3)

Z

j2 +

The group action (3.2) as one can easily verify is Poissonian, leaving the symplectic structure (3.1) invariant. Thus if a one parametric subgroup fexp(a()t) : a() 2 A+ (); t 2 Rg acts on M;the corresponding Hamiltonian function comes as follows: Ha = ,res2D0 Sp(Q|  ,1 Fa()) := ,2 < l(F; Q; ); a() >0 ; (3.4) where l(F; Q; ) := 21 Q|  ,1 F (3.5) is the momentum mapping [1,2] and <
;
>r r 2 Z, is a scalar product on A(; ,1) de ned by the expression: < l(); a() >r := res2D0 ,r Sp(l()a()): (3.6) It is easy to verify that the momentum mapping l : M ! A+ () de ned by (3.5) is equivariant [1], that is the diagram

M ? !l A ?? + () g()y yAdg,1 () M !l A+ () 5

(3.7)

is commutative for all g() 2 A+ (); meaning [1] that the loop group A+ () action on M is Hamiltonian. De ne now a Lie algebras homomorphism  : A+ () ! G+ ()  2 A+ ()  + R; (3.8) where for any a() 2 A+ ()

(a)() := 2 a()  a(0) (3.9) 21 +       with + = 00 10 ; , = 01 00 and 0 = 10 0,1 being a sl(2; R) matrix basis.. It is veri ed that the mapping (3.8) is a homomorphism and the image A+ () := G+ () constitutes a Lie algebra over R. Thus there exists a loop group G+ () whose Lie algebra coincides with this Lie algebra G+ (). Thereby one can de ne now another loop group G+ (),action on M de ned by the formulas (3.2) but with an element g() 2 A+ () replaced by an element g() 2 G+ (); where  : A+ () ! G+ () is the corresponding to the mapping (3.8) loop groups homomorphism. Therefore, similarly to (3.4) one nds a momentum mapping l : M ! G+ () with respect to the modi ed loop group  action G+ ()  M ! M equivalent to that of A+ ()  M ! M: A simple calculation yields (0) + l (F; Q; ) = l(F; Q; ) + l12  ; (3.10) P where by de nition, l := j 2 + l(j ),(j +1) .When deriving (3.10) we based on the Hamiltonian function expression Ha = ,2 < l (F; Q; ); (a)() >,2 (3.11) generated by a one parametric subgroup fexp(a()t) 2 G+ () : a() 2 A+ (); t 2 R g and made use of the properties Sp(  ) = 1; Sp(, + ) = 0 = Sp(+ , ) for the dual bi-orthogonal basis f , 0 g 2 sl (2; R): Notice now that the element  := 2 + , 2 2 G  (; ,1 ) is an in nitesimal character of the Lie subalgebra G+ (); where by de nition G (; ,1) := G+ ()  G, () and

Z

< ; [G+(); G+ ()] >,2 = 0 =< ; G,() >,2 : (3.12) Owing to the property (3.12) and AKS-theorem [7-9], the extended momentum mapping S (F; Q; ) := 2 + , 2 + l (F; Q; ) (3.13) generates on the manifold M an involutive with respect to (3.1) invariants

j 2 D(M ); j = ,1; n; via the expression: n X (3.14) det S (F; Q; ) = ,2 +  ,1 + 0 +  , j ; j j =1 6

where we have put for de niteness

:= diagf j 2 R=f0g : j = 1; ng; Q := Fh;  1 ; F := q1; q2 ; :::; qn 2 M . As a result of simple h = 01 , n;2 0 p1; p2 ; :::; pn calculation one nds from (3.14) that

j = , 12 p2j + 14 < q; q > qj2 , < q; q > 2j qj2 + (3.15) n 1 X 2 4 k6=j =1(pj qk , pk qj ) =( j , k );

where j = 1; n; and <
;
> is the usual scalar product in Rn: The corresponding symplectic structure (3.1) turns into the following canonical one: !(2)(F; Q) = 2!(2) (q; p); where !(2) (q; p) := Pnj=1 dpj ^ dqj : Thus all Hamiltonian ows generated by invariants (3.15) on the space M ' T  (Rn) are Liouville-Arnold integrable by quadratures since f Pj ; k g = 0 for all j; k = ,1; n: In particular for the Hamiltonian function H := nj=1 j j the corresponding dynamical system on T  (Rn) is given as follows:

dqj =dx = pj ; dpj =dx + 2j qj , j qj < q; q >=

(3.16)

qj (< q; q > ,3=4 < q; q >3 ); where j = 1; n: Similar to (3.16) oscillatory equations constrained to live on the cotangent space T  (Sn,1) to the unit sphere Sn,1 = fq 2 Rn :< q; q >= 1g were for the rst time derived and studied in detail in [5,14], having based exclusively on the algebraic-geometric techniques [15]. Later on these results were rederived in [5,16] from the Lie-algebraic viewpoint [6]. As was shown in [17] by means of direct calculation, the extended momentum mapping (3.13) satis es the following dynamical r,matrix identity: fS (q; p; ); S (q; p; )g = [r12(; ); S (q; p; ) I] , [r21(; ); I S (q; p; )]; (3.17) where r21(; ) := r12(; ) and r12(; ) = P=( , ) , (< q; q > , , ), + ; (3.18) Px y := y x for any x; y 2 R2 and all  6=  2 C : There is an important problem of deriving this r,matrix (3.18) from the pure Lie-algebraic viewpoint as it was done in [17] subject to the Calogero type models.

4. Acknowledgements.

The author acknowledges an AGH-local grant he bene ted at the very beginning of work upon the problem. He is also grateful to his students for discussion of many aspects of the reduction structures considered in the article. 7

References

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[16] Prykarpatsky A., Hentosh O., Blackmore D. The nite-dimensional Mosertype reduction of Modi ed Boussinesq and super-Korteweg-de Vries Hamiltonian Systems via the gradient-holonomic algorithm and dual moment maps. Part 1. J. Nonl. Math. Phys., 4(3-4), 455-469 (1997) [17] Avan J., Babelon O., Talon M. Construction of the classical R-matrices for the Toda and Calogero models. Preprint LPTHE University Paris VI, CNRR UA 280, PAR IPTHE 93-31, 22p (1993) [18] Prykarpatsky A.K. The nonabeliean Liouville-Arnold integrability problem: a symplectic approach. J. Nonl. Math. Physics, 6(4), 384-410 (1999)

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